Hausdorff dimension and conformal measures of Feigenbaum Julia sets

Abstract

We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the “hairiness phenomenon”, there exist many Feigenbaum Julia sets J ( f ) J(f) whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigenbaum Julia set, the Poincaré critical exponent δ c r \delta _{\mathrm {cr}} is equal to the hyperbolic dimension H D h y p ( J ( f ) ) \mathrm {HD}_{\mathrm {hyp}}(J(f)) . Moreover, if area ⁡ J ( f ) = 0 \operatorname {area} J(f)=0 , then HD h y p ⁡ ( J ( f ) ) = HD ⁡ ( J ( f ) ) \operatorname {HD}_{\mathrm {hyp}} (J(f))=\operatorname {HD}(J(f)) . In the stationary case, the last statement can be reversed: if area ⁡ J ( f ) > 0 \operatorname {area} J(f)> 0 , then HD h y p ⁡ ( J ( f ) ) > 2 \operatorname {HD}_{\mathrm {hyp}} (J(f))> 2 . We also give a new construction of conformal measures on J ( f ) J(f) that implies that they exist for any δ ∈ [ δ c r , ∞ ) \delta \in [\delta _{\mathrm {cr}}, \infty ) , and analyze their scaling and dissipativity/conservativity properties.